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We consider the hyperelliptic handlebody group on a closed surface of genus$g$. This is the subgroup of the mapping class group on a closed surface ofgenus $g$ consisting of isotopy classes of homeomorphisms on the surface thatcommute with some fixed hyperelliptic involution and that extend tohomeomorphisms on the handlebody. We prove that the logarithm of the minimaldilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in thehyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This meansthat the asymptotic behavior of the minimal pseudo-Anosov dilatation of thesubgroup of genus $g$ in question is the same as that of the ambient mappingclass group of genus $g$. We also determine finite presentations of thehyperelliptic handlebody groups.

The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups